Since they are rather important and quite accessible, we repeat the general theoretical facts concerning weights, moments and polynomials pertaining to fourth order differential equations. We then briefly discuss the squares of the differential equations of the second order, giving a number of easily derived examples of fourth order problems. This is followed by three new orthogonal polynomial sets satisfying fourth order differential equations, but which do not satisfy second order differential equations.

Orthogonal Polynomials Satisfying Fourth Order Differential Equations

Boundary problems for ordinary differential equations with parameter in the boundary conditions

Sturm theory is extended to the equationfor 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditionsandOscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C24388CFF23FEF3894645CBB5A460D9F/S0013091500018691a.pdf/div-class-title-sturm-liouville-problems-with-eigenparameter-dependent-boundary-conditions-div.pdf

Sturm–Liouville problems with eigenparameter dependent boundary conditions

Software is described for the Sturm-Liouville eigenproblem. Eigenvalues, eigenfunctions, and spectral density functions can be estimated with global error control. The method of approximating the coefficients forms the mathematical basis. The underlying algorithms are briefly described, and several examples are presented.

Mathematical software for Sturm-Liouville problems

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrodinger operators on [a , ∞), a ∈ ℝ, with a regular finite end point a and the case of Schrodinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrodinger operators on (a , ∞) with a potential strongly singular at the end point a . We focus on situations where the potential is so singular that the associated maximally defined Schrodinger operator is self-adjoint (equivalently, the associated minimally defined Schrodinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a . For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrodinger operators. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On spectral theory for Schrödinger operators with strongly singular potentials

In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions

In this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500012312

Charles T. Fulton

Papers

Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions

Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions

Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions